科恩扩展中的滤门格尔实数集

Pub Date : 2024-07-15 DOI:10.1002/malq.202300008
Hang Zhang, Shuguo Zhang
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We show that in the Cohen model there exists such <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$\\mathfrak {d}$</annotation>\n </semantics></math> that is not Hurewicz in the <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-Cohen model where <span></span><math>\n <semantics>\n <mrow>\n <mi>κ</mi>\n <mo>&gt;</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>$\\kappa &amp;gt;\\omega _{1}$</annotation>\n </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n <mo>&lt;</mo>\n <mi>d</mi>\n </mrow>\n <annotation>$\\mathfrak {b}&amp;lt;\\mathfrak {d}$</annotation>\n </semantics></math>. We also study the filter <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math> generated by the set of mutually Cohen reals in the <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-Cohen model. We prove that <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n <mrow>\n <mo>(</mo>\n <mi>F</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>$\\mathfrak {b}(\\mathcal {F})=\\omega _{1}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>(</mo>\n <mi>F</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>κ</mi>\n </mrow>\n <annotation>$\\mathfrak {d}(\\mathcal {F})=\\kappa$</annotation>\n </semantics></math> and every <span></span><math>\n <semantics>\n <msup>\n <mo>≤</mo>\n <mo>∗</mo>\n </msup>\n <annotation>$\\mathord {\\le ^{*}}$</annotation>\n </semantics></math>-dominating family in the ground model is <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math>-unbounded in extension. Two questions are posed.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Filter-Menger set of reals in Cohen extensions\",\"authors\":\"Hang Zhang,&nbsp;Shuguo Zhang\",\"doi\":\"10.1002/malq.202300008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that for every ultrafilter <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$\\\\mathcal {U}$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <mi>ω</mi>\\n <annotation>$\\\\omega$</annotation>\\n </semantics></math> there exists a filter <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <msup>\\n <mn>2</mn>\\n <mrow>\\n <mo>&lt;</mo>\\n <mi>ω</mi>\\n </mrow>\\n </msup>\\n <annotation>$2^{&amp;lt;\\\\omega }$</annotation>\\n </semantics></math> which is <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$\\\\mathcal {U}$</annotation>\\n </semantics></math>-Menger and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>χ</mi>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>b</mi>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\chi (\\\\mathcal {F})=\\\\mathfrak {b}(\\\\mathcal {U})$</annotation>\\n </semantics></math>. We show that in the Cohen model there exists such <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$\\\\mathfrak {d}$</annotation>\\n </semantics></math> that is not Hurewicz in the <span></span><math>\\n <semantics>\\n <mi>κ</mi>\\n <annotation>$\\\\kappa$</annotation>\\n </semantics></math>-Cohen model where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>κ</mi>\\n <mo>&gt;</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\kappa &amp;gt;\\\\omega _{1}$</annotation>\\n </semantics></math> is uncountable regular. 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We prove that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>b</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\mathfrak {b}(\\\\mathcal {F})=\\\\omega _{1}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>$\\\\mathfrak {d}(\\\\mathcal {F})=\\\\kappa$</annotation>\\n </semantics></math> and every <span></span><math>\\n <semantics>\\n <msup>\\n <mo>≤</mo>\\n <mo>∗</mo>\\n </msup>\\n <annotation>$\\\\mathord {\\\\le ^{*}}$</annotation>\\n </semantics></math>-dominating family in the ground model is <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math>-unbounded in extension. 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引用次数: 0

摘要

我们证明,对于上的每一个超滤波器,都存在一个滤波器,它是-门格尔和 。我们用 Nyikos [10] 的构造证明,在科恩模型中存在这样的高滤波器。这回答了达斯的一个问题[2, 问题 7]。我们证明,在-科恩模型中,存在一个不可数正则表达式的门格尔滤波器的特征不是胡勒维茨。这表明对埃尔南德斯-古铁雷斯和塞普蒂奇[3, 问题 2.8]问题的肯定回答与 .我们还研究了-科恩模型中互为科恩有数集所产生的滤波器。我们证明了地面模型中的 和 以及 每个主族在广延上都是无界的。我们提出了两个问题。
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Filter-Menger set of reals in Cohen extensions

We prove that for every ultrafilter U $\mathcal {U}$ on ω $\omega$ there exists a filter F $\mathcal {F}$ on 2 < ω $2^{&lt;\omega }$ which is U $\mathcal {U}$ -Menger and χ ( F ) = b ( U ) $\chi (\mathcal {F})=\mathfrak {b}(\mathcal {U})$ . We show that in the Cohen model there exists such F $\mathcal {F}$ which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character d $\mathfrak {d}$ that is not Hurewicz in the κ $\kappa$ -Cohen model where κ > ω 1 $\kappa &gt;\omega _{1}$ is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with b < d $\mathfrak {b}&lt;\mathfrak {d}$ . We also study the filter F $\mathcal {F}$ generated by the set of mutually Cohen reals in the κ $\kappa$ -Cohen model. We prove that b ( F ) = ω 1 $\mathfrak {b}(\mathcal {F})=\omega _{1}$ and d ( F ) = κ $\mathfrak {d}(\mathcal {F})=\kappa$ and every $\mathord {\le ^{*}}$ -dominating family in the ground model is F $\mathcal {F}$ -unbounded in extension. Two questions are posed.

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